The Rate of Convergence of Randomly Indexed Sums of Independent Random Variables to a Stable Law

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ANNALES UNIVERSITATIS MARIAE CURIE-SKLODOWSKA LUBLIN-POLONIA

VOL. XLV1, 6__________________________SECTIOA____________________________________1992

Andrzej KRAJKA Zdzislaw RYCHLIK (Lublin)

The Rate of Convergence of Randomly Indexed Sums ofIndependent Random Variables to a Stable Law

Abstract. This paperpresents uniformand nonuniform rates of convergence ofrandomly indexed sums of independentrandom variables to astable law. The presentedresults extendto the caseof randomlyindexedsums those given by A. Krajka andZ. Rychlik (1933).

1. Introduction. Let {Xn,n > 1} be asequence of independent random vari ables whosedistribution functions {F„,n > 1} belong to thedomainof attraction ofa stable lawof exponenta, 0 < a < 2. Then, undersomeadditional assumptions, there exist constants {r„,n > 1) and {<„,« > 1} such that — r>)/<„ converges to a stablelaw, as n —> oo. Thespeedof this convergence is given in [4]. The main results of [4], Theorem 1 and 2 are repeated here. On the base of this results we derive uniform and nonuniform rates of convergence of randomly indexedsums to a stablelaw. Uniform rate ofconvergence is presentedin Theorem 3,while nonuniform in Theorem 4. Itshould be mentioned here that wedo not assumeanything concern ing the interdependence betweenrandomindices and randomvariables {X„,n > 1}.

Theorems 3 and 4 seem tobenew inthe context ofstable convergence.

We close this section with some notations. Let {<frn,n > 1} be the sequence of the characteristic functionsof{Xn, n >1}, and let {c,¡¿, j > 1}, t = 1,2, be sequences of nonnegative numbers such that Cij + cjj > 0, j > 1. Define

M*) = 1 - Fj(x) +F/-x)-(cjj+c2i>)(z-° A 1) , d,(x) = 1- F/x)- F>(-x) -(c1(, - ca,,Xx-° A 1) ,

n

= D„(x)= ^d>(x),

>=i >«=i

Hn(x) = ¿ |Mx)| , D„(x) =¿ |d/x)| ,

>■1 j=i

Here, and in what follows, x V y= max{x,j/} and x A y = min{x, j/). Let us put Í - /“ u~° cos(u)du , if a € (0,1),

*ej = I u-0,sin(u)du , e2 =< 1 , if a = 1, ,

° I /o°° ““"t1 ~ cos(u))du , >f a 6 (1,2),

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44 A. Krajka ,Z. Rychlik

zoo zoo

bltj =J (1 - cos(fx))d/»>(x) or = sm(tx)hj(x)dx , /o°° »™(tx)ddj(x) , for a € (0,1)

/0°°(l- cos(tx))dj(x)dx , for a € (1,2),

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°1 J = (cl,> + C2,>)el , a2,j = (cl,j + c2,>)e2 ,

= jf (! ~ Fj(x) - Fj(-x))dx + dj(x)dx-(cltj-c2,>)7 ,

r. „/#»>+' 5Xia».<lo«(<“/i“-i) + °2.>k’SUj') , >fa= 1,

I 0 , otherwise , j > 1,

n

7= lim V j-1—Inn) (Euler’s constant), n—»oo

•«1

n n

Cl,« = , Cj,B = 5"^ c2,ji

>=1 2-1

<“ — (Ci,n+ Ca,«)ei , ?o = 0, n > 1 Throughoutthe paper, we assumethat for some real number A

A= lim(Ci,„ — C2,„)e2/(Ci>n+C2,»)ei , (2) lim <„ = +oo ,

andfor j>1,EXj = 0, iff it exists.

Let Ga,\( • ) denote thestable lawwith the characteristic function (3)

Letus define

,{t}= i exP{-|<l°(1 + ‘^‘P1 (0)} , if <*#1, 1 exp{—|f|(l+ »Asign (<)ln|t|)} , if a = 1.

A,(i) = |P[S„ < x<„] - G<,,a(x)| ,

where S^E-^-r,).

Let {7Vn,n > 1} be a sequence of positiveinteger-valuedrandom variables such that

(4) —*(> asn-4OO,

where £ is a positive random variable independent of {2V„,n > 1} and {An,n > 1}.

Let usdefine

T„ = max{t: < £<“} A(2V„,x) = \P(SN. < xiN.) -Go,a(x)| ,

A(T„,x)= |P(S;v. < x<tr) - G<«,a(z)I >

A(n, (, x) = \P(SN, < x(1/o<„) - GOtX(x)| , n > 1 .

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The Rateof Convergence ofRandomlyIndexed Sums... 45 In what follows C,denotes the positive constant whichmay only be dependent on a and A.

2. The rates of convergence to a stable law. In [4]it isproved:

Theorem 1. Let {X„,n > 1} be a sequence of independent random variables, and let {e„,n > 1} be a sequences ofpositive numbers such that for every0 < t < rj andn > 1,

4max{f°'(|a,J + |a2j|)+ |ti,j(t)| + |^,>(t)|} < 1 , if a / 1 ,

(5) 4max{t(|a1,J + |a3,>(logt)2| + |/t>|) + + |*>2.j(.<)|} < 1 . »/<*=!,

where rj is some positive number.

Assume

d(i)= sup{7Z„(x)VDb(x)}<~° =o{x~a) as x -> oo ,

(7) "

and supx“t?(x) < oo ,

X

and, in addition, ifa— 1 then

(8) max|iw(t)| |ln(f)| <oo ,

for0 <t < rj.

(a) . If {h„,n > 1} and {d„,n > 1} ore sequences of uniformly ultimately mono

tone functions on [0, oo) and 0< a < 1, then

(9) supA„(x) < y x|Kn(x)|dx + <“* y |£>„(x)|dx+

+ /‘0°z-1{|/fn(x)| + |DB(x)|}dH-£B+<(2“Al)} =U,(n) .

(b) . If f£°t?(x)dx < oo , then for a = 1 (10)

supA„(x) < CpB2 jf x|ff„(z)|dx +fB’y X2|PB(x)| dx+

+<T1 / l + Pn(x)| ]dx + £„ +iB2(log<„)2} = Ui(n) .

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46 ^A.^Krajka^{, Z.}^Rychlik

(c) . If (b) holds and{/i„,n> 1} w o sequence ofuniformly ultimately monotone functions on [0, oo), then fora = 1

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supAn(x)< c{q~2 [ x\Hn(x)\dx +q~3 i x2|D„(x)|dx+

z '■Jo .Jo

+ [ x~i\Hn(x')\dx+q~1 i |D„(x)|dx + e„ + <2(log?„)2j = Uc(n) .

An Jf„ 1

(d) . If > 1} M a sequence ofuniformly ultimately monotonefunctions, then for 1 < a <2

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supA„(x) <c{q~2 i x\Hn(x)\dx+q~3 i x2|P„(x)| dx+

* 1 Jq Jq

+ [ x~'\Hn(x)\dxi |D„(x)|dx+e„ + ?~2j = Ud(ji) .

•let •!<:» }

(e) . For 1 < a<2

(13) supAn(x) < C|c~2 y x\H„(x)\dx + q~3 f x2|Dn(x)| dx+

+ /°°(|/fn(x)| * |P„(x)|)dx + en + <’} =Ut(n) .

J<„ '

Theorem 2. Let {X„,n > 1} be a sequence of independent random variables satisfying the~assumptions (5)-(8) of Theorem 1 Iffor every n > 1, the functions Hn(x) andL„(x) are ultimately monotone on [0,oo) and 1 <a < 2, then for every xe R

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(1+ |x|)A„(x) < Cic’1 /*(1 + |ln(x/<B)|)[0„(x)+ffB(x)]dx+

+-ffn(in)+ <3 y x2Z>„(x)dx + q~2 y x/f„(x)dx +en + ?“-2j =

The following theorem presents the uniform rate of convergence of randomly indexed sums of independent nonidentically distributed randomvariables to a stable law.

Theorem 3. Let {X^n > 1} be a sequence of independent random variables satisfying the assumptions (5)-(8) of Theorem 1 with hn(x) > 0, d„(x) > 0, n > 1, such that for every 0 < p < q, p,q € N,

(15) sup|ff4(x)-ff,(x)||x“Vl|<C(i4“-<;), H0(x) = 0 ,

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The Rate of Convergence of RandomlyIndexed Suma... 47 and, in case a

= 1,

D,(x)|<fc< C(<,(ln(<,) V 1) - <,(ln(<,)V 1)) .

Assume, for some t € {«,b,c,d,e},the assumption (i) of Theorem 1 holds and for some nonincreasing sequence {¿(n), n > 1} Ui(n) <?(n) —♦ 0 as n—♦ oo, where

(«(«) , »/q€(0,1J, b(n),/2 , if a €(1,2).

If{Nn,n> 1} is a sequence ofpositive integer-valued random variables such that (16)

l<Nn/<T. ~ 1| > Ce(n)] < C?(n) , n > 1 , if a/ 1

P( |Qv„(ln(<N.) V1) - <T.(ln(<T.)V1)| > C£(n)<r„) < C?(n) , n > 1, ifa = I and

(17) P(a<(iZ)< C/c(n)] <C?(n) , n > 1 ,

where _OO

°n(U) = YSlUkV-'ltf < < tf+1) , n> 1 Jhxl

then

(18) sup A(r„, x) <Ce(n) , n>l, z

and

(19) supA(7V„,x) < C?(n) , n > 1 . If, additionally, there exists a constant Co such that

sup(ci,*+i +c2,*+i)<f“^(i)_1 < Co , then

(20) supA(n,(,x) < C?(n) , n > 1 .

Let us observe,that assumptions h„(x) >0, dn(x)> 0, n > 1 in Theorem3 may be Replacedby the followingone:

There exists a positive number p such that the sequence {q£Ui(k),k > 1} is nondecreasing.

The nonuniform bounds are the following

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48 ^A.^Krajka^,^Z.^Rychlik

Theorem 4. Let 1 < a < 2 be given and let {X„,n > 1} be a sequence of independent random variables satisfying the assumptions (5)-(8) ofTheorem 1, and (15). Assume, for some nonincreasing sequence (c(n),n > 1} U{n) < e*/2(n) —» 0 as n —♦oo. If {N„,n >1} is a sequence of positive integer-valued random variables such that for every |x| > 1

Pl 1^.At. - H > clC(n)|x|“-*] < Ce1/2(n)/(l +|x|) ,

<1 ,

P|1 - c,< < 1 + Ce’/’WIxl“-1]< C?/2(»)/(l +|x|) ,

»/e(n)|x|“-1 >1 , and

(22) P[<\(t/)< C/e’/’in)] < Ce<a+",Wa-'»(n) , i/e^lxl“-1 < 1 , where

°<(V)

⁼^{¿(^(fcM-’}^/Af ^<a

: < <?+1 ) ,

n >1 ,

t=l

then

(23) A(T.,«)<Ce1/,(»)/(l +|x|), n > 1 , and

(24) A(JV„x)-< C?/2(n)/(l + |x|) , n> 1 .

If, additionally, in case e(n)|x|“_1 < 1 there exists a constantcq such that supAiji+i +c2,t+i)ifQ'tZ(fc)-1 < co ,

then for every x € H

(25) A(n,(, x) < Cex'\n)l{l + |x|) , n >1 .

3. Auxiliary lemmas and proofs. Intheproofs of Theorems 1 and 2 weneed thefollowingthree Lemmas. Lemma 1 isproved in [4],Lemma2 extends Lemma 10 [6] to thecase a 2 and Lemma 3 follows from[2] and [8].

Lemma1. For every p>0, 0<o<2 and Aj e [-1,1], A2 e [-1,1]

(26) sup|Go,Al(x) -Co>Al(x)| < (T(l + l/«)/(»a))|A, - A2|

X

(27)

(1+ l*|)|Ga,Al(x) - C?o,A,(x)| < 640(2T((2a - l)/a) + (2+ l/«)r(l - l/o))

|A1-A2|/x (t/1 < a <2)

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The Rate of Convergence of RandomlyIndexed Sums... 49 (28) sup|Go,A,(i +p) - Go,a,(x)| <(Pr(l/«)/(»Q))Al

r

(29) |Go,a,(x+ p) - G„,x,(ar)i < T(a)j |x+ p| ° -k|“®| (for 0 < a < 2, a ± 1)

(30) SUpIGa.Ajpxi-Go.Ajx)! < « X

¿(p° V p_<* +|Ai|)|l — p"A p-o| , i/o^l, ill -p~* ApKpAp-1 + |Ai| |7-2«i-l)|)+

+J|Ai| llnpl » */<* = !.

anJ

(31) |C«,.Al(px) - Go,m(x)| < r(a)|*|-|p- V Pa - l|/x . where

Lemma 2. Let X be a random variable. IfY is anonnegative random variable such that forsome £j > 0 , ej > 0 and 0 < a < 2

P[ |K° -1| >£l] < e, , then

(32) where

sup |P[X <xK] -G„,a(x)| <sup|P(X< xl- G«,*(x)| +e2 +C,£i ,

Z z

c i ¿(2+1*0 +1 , ¿/a#l,

* U(3 + |A|(7-2«(-l) + l)) + l , «/a = 1 .

J/ei|x|“_l < 1/2, 1 < a < 2 and

P[ -1| >c1kr~1] <«»/(l+

kl) ,

fAen (33)

|P[X < xK] -Go,x(x)| < |P(X <x(l - siki“-*)*'«]

- Go>a(x(1 - £1|x|“'1)1/q)I V |P[X < x(l +£1|x|“-1),'“l

-Ga,A(x(l + e, ki“-1)1/“)!+ 4e,r(a)/(a(l + |x|))+d/(l+kl) •

. Proof. Assume 0 < £i < 1, as in case ffi > 1 (32) is obvious. One can easily observethat ifx > 0, then

P[A < x(l - fi)1/o] -c2 < Ppf< xK]< P(X < x(l+ci)1,“l + ej •

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50 ^A.^Krajka^{, Z.}^Rychlik On the other handif x < 0, then

P[X <x(l +c,)1/o]-£2 < P[X<xK] < P[X < x(l — ei)1/o] + e2 • Hence

sup|P[X <xY] - Gq,a(x)| < sup|P[X < x)- G„,a(x)|+

X X

+sup|Gq,a(x(1 -ei)1/<*)-Ga,A(x)| Vsup|Go,A(;r(l + ei)1/o)- Go,a(x)| + Cj .

X X

Thusby Lemma 1 one can easily get (32). Onthe otherhand, using the inequalities obtainedabove with £j replaced by £j|xj“-1,we get

|P[X < xY] - GOlX(x)\ < |P(X < x(l -£,|i|°-1)1/“] -G„,a(x(1

V ¡P(X < x(l + c,«-1)1'“] - G«,a(x(1+ ii|x|o"1)1/“)| + |G„,a(x)-

-Go,a(x(1 - ci |x|"-1 )1/o )|v|Go,a(x) -G„,a(x(1 + c,|x|a,-1)1/o,)|+ c2/(l + |x|) Thus Lemma1 ends the proof of (33).

Lemma 3. Let {X„,n > 1} be a sequence of independent random variables.

Then, for every x > 0,

P[ |S„| >x«,] <P[ max IS,/«| > xj <

l<i<n

exp{—Gj((x - x0)/(2x0))(ln2)/(ln3)} + H„(x<„)+x~a and

max{l -Ga>A(x),GOiA(x)} < G|x|-“ , where Ci = (ln3)/2 and xq is defined by

P[|S„|>x0]<^.

Proof ofTheorem 3. At first assumethat a 1. Using the fact that £ is independent of{Xn,n > 1}, we get

oo

|P[5r. < xqj- Go,a(x)|< £p[T„ = ¿] |P[S* < xq]- Go,a(x)| <

*=i

< P[o„(iZ)< C/e(n)] + £ PlT" = *1 lPlS* < »«’-I ~ G-.a(»)I

*€D(»)

where here, and inwhat follows D(n)= {fc : Uj(Jfc) <c(n)/G}. Thus by Theorem 1, i34) (¿’[•St. < *<T„] - GO|a(x)| <CZ(n) , n > 1 .

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Tbc Rateof Convergence of Randomly Indexed Sums... 51 Let In = {Jfc > 1: Then,according to (16)

(35) P[JV. i /«J < C?(n).

Let us put

A„(x) =[maxS* < x«rJ , P„(x)« [min St < x«t„] , n> 1 .

*€/« *€f«

Then, by (35)

(36) P[A»(x)l - CZ(n) < P[SN. < x<T.] < P[Bn(*)l +C?(n) , n > 1 . On the other hand

(37) P[A.(x)] < P[SN. < x<t.] < PtfMx)), n> 1 , so that (34), (36) and (37) yield

(38) sup A(T„, x) < sup{P[PM(x)l - P(A„(x)]} + C?(n) , n > 1 .

x x

Thus (38) yields (18) ifitis shownthat

(39) sup{P[P„(x)] -P[A„(x)]} < CZ(n) , n > 1 .

x

But (39) is bounded, from above, by (40)

P[crn(CT) < C/?(n)]+sup V = *1 lpl min 5< < max <*<*1I>

’ *«£?») ’€/(‘>n)

where

/(*,») = {«•: tf(l -Ce(n))< < < (1 +Ce(n))} . Furthermore, for every p G I(k,n), wehave

(41) P[. min . Si <x<*l - Pi max Si < ^xq] t€/(fc,n) /€/(*, n)

=P[ max 5< > XQ > Sp] — P[ min S< < xq < S.l .

i€.l(k,n) i€/(A,n)

We give the evaluation of the last term in (41) only, since the second one can be evaluatedsimilarly. Letus put p = p(k,n) = min{t : t € f(fc,n)} andq = q(Jc,n) = maxft:» G i(fc,n)}. Then, by Theorem 1 and Fubini’s theorem, we get

(42) P[. min S, > x<* > Sp] = / Pi.. •P,{(*i

»€/(«,n) J

J f

*<*/<;,- max *</<> 12^{Zi^r -}AfJ^f+i " •F»

i=F+i i-1

(X,+1,. • •, 2,) < CUi(p) + I |Go,a(2Q/<,) - G«,>(«*/?,-

i

“,<>< 52 Z<Af)MP>+1••• Pf(*F+i»•••»««) ^

*iasF+l

<CUi(n)+P( max sup|Go,a(x) -Go>a(x- (S>- S,)/?,)!) . P<J<1 x

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52 ^A.^Krajk*^{, Z.}^Rychlik Furthermore,by our assumptions, wehave

(43) < (1- Ce(n))-3'aCUi(k) <£e(n) .

For example, in case » =a, (43) follows from inequalities

f ‘

^(x-¹^-^{x<J(l -}^ce(n))-’^/^‘’)|ff^p(x)|^{dx <}⁰^,

A»(l-€<(»))*/•

/ (* 1 --«(«)) 1/")I-dp(i)Idx <0 , Ak(l—ee<»))1/«

i

e, < e» + |52 e* + (<" - <p )A P < e* +

Ce(n) ,

and,for each x > 0,

|2i,(x)| < |fft(x)| , |Dp(x)| < |Dt(x)| .

If 0 <a< 1, thenwe apply q — p timesLemma1 and get

I = E(max sup |Go,a(x) - Ga,x(x - (Sj -

SP )Apl) <

FSJSf *

<£ E |X,IA|X,|<<P)A,+ E P[|X>|XP]<

>=p+i >=p+i

< - j T

^X^d(ff, ^-^H^r^)(x)/,r^-^{(if“ -}^?p^“⁾^£^X^dx-«/ip+

+ (if(<P)'-Hp(<p))+«-«;)/<;< 1

<jp (ff,(x) -j?p(x))dx/<p+ c(<;- <;>/<;<

On theother hand, if 1 < a < 2, then Holder’s inequality and Lemma 1 (also

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The Rate of Convergence ofRandomly Indexed Sums. 53 applied q — ptimes)give

I =E(max sup|Ga,x(x) - Ga,\(x - (Sj - Sp)/ft>|) <

?<■><« *

< £| £ XyJTO<<,)|A,+ £ Pl |X>| > <,]<

W+l

f . 1 x 1/2

< { £ fW<„)’/(PGI < ip)} + £ E\Xi\M\Xi\ > <p)+

>=p+l >=p+l

>=p+l

{- £' x d(H,* -

h

,

m

/<} - (SQ - <;) £ x2 «b-/<j}’/2+

+ {- J°° xd(H, -K,)(x)A,-(<; -^^xdx-A,}172- - r x d(H, - H,)(x)/q, - - <?) r x dx~a/,p <

Js, '

< {-(#,(<,)- #,(<,)) +«(< - tf) £’ X1"“ &/<*+

+ 2/ X^HM~ Hr{x\)dxlci} ' + {J (Hg-Hr)(x)dx/<;P+

+(H^r) - iw,))+ ««■ -<;> fx- <w<U1/2+ + (H,(<p) -H,(q,» +«(<? - S“) /°°x~° dx/qp+

+ [°°(H,- Hp)(x)dx/<;p < C(c® - <)1/2/?F“/2 . Jg.

Thus, bythe estimations given above, we obtain

(44) I< CZ(n).

Hence, in case a /1, (18) is aconsequence of (38)-(44).

If a= 1, then

(45) kjv.(ln(iN.) V 1) - ir.(ln(iT„) V 1)|/<t. > |iN./iT. - H so that

/*(*:, n) = {«: Q(ln(?t)V 1- Ce(n)) < ft(/n(ft)V 1)< <*(/n(<*) V 1 + Ce(n))} C C {» : <t(l -Ce(n)) < ft <q(1+ Cc(n))} . .

Thus the estimations(34)-(43) also holdwithI(k,n) and/„ replaced by I*(k,n) and

= {t > 1:k* ln(<*) V1 - <r. M<r„ V 1| < Ce(n)iT.} ,

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54 A. Kr&jka, Z.Rychlik

respectively. Furthermore the term I, in the case a= 1,can be estimated asfollows

I< C<st - Ce(") +{ f

!D«() “ A’i)! dx+

>r+>

+ (<, - <,)(! + 7) + («, - <p)ln(<p) + C, ln(?i/<,)}/<,<

<Ce(n) + ln(i,/t,) < Ce(n).

Thus,inthe case a = 1 (44) alsoholds, so that (18) holdsfor every 0 < a <2.

Now (19) follows from (18), (16) and Lemma 2. In the casea = 1 we also use (45). On theother hand if

sup{(ci,*+i +c2,t+1X^"Cfi(^)-1} < co ,

k

then

P[ ICtf/tf.- 1| > coC-*?(n)) = £ P«<?> $ +coC-*Z(nXf < tf+1] <

k=l

< > ?*+(ci,k+i+c2lk+1)C~1Efj(fc)"1?(n),c* < < ?*+1] <

*=i

< Pk»(tZ) < C/?(n)] < C?(n) .

Thus (20) is also a consequence of (18), (16) and Lemma 2, so that the proof of Theorem 3iscompleted.

Proof of Theorem 4. At first letus consider the case c(n)|x|“-1 < 1 . Since, in this casefor|x| > 1

e(“+»)/(»(«-D)(n) =e1/l(n)e1/(“-,)(n)<«’/’(nJlxl"1 , sothat (22) may bereplacedby

P[ac(tf) < C-/?/J(n)]< + |x|) . Now, putting

/.(*) = {* > 1=k? - <?.| <cl£(n)|x|“-\?J ,

/(i,n,x) = {«■> 1: tf(l - c,e(n)|x|“-1) < (1 + dc^lxl“-1)}

and proceeding step by stepas in the proofof Theorem 3 we obtain

A(T„,x)< C?/2(n)/(l+|x|)+P(. max Si >x<* > Sp]+P[ min Si < xq < SJ

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The Rateof Convergence ofRandomly IndexedSums... 55 Furthermore, the sum of two last terms, by Theorem 2, may be bounded by

" G°^x “ (Sj - * 1 + |x| ^Kl<l

-cTVm +£( <T< Ka(1- i xj(|xj<<,)/<,)-c<u*)|)+

i=>p+i

+ £(max |g„,a(x- £ XJdXH < «)/«) -G„.a(x- (S,- S,)/c,)|) <

“ >=p+i

-cTTkT+ c(1 + l*,“+,)',[{ i EX?idXd<s,)/^y/\

1 >=F+1

+ £ WAIXJ> <,)/<,] + P( max |S, - Spl/i, > |x|/4]+

i=p+i

+ P[max | XJ[|X,|< <,]/<,| > |x|/4] + P-J-1 i=P+i

=G?/’(n)/(l+|x|)+G(1 + |x|“+1 )-*[/,+ /,]+/,+/<, where p =p(x) = min I(k,n,x), q = g(x) = maxI(k,n,x).

The estimation of A and I? are given in the proof of Theorem 3. On the other hand, the estimations of Zj and I4 one can get from Lemma 3. Thus (23) follows.

Inequalities(24) and (25) followfrom (23) and the assumptions similarly as in the proof of Theorem 3.

Assume e(n)|x|<*~1 > 1. Then

(46) (1 - Go,a(|x|)) V Go>a(-|x|) < G|x|-° < Ge(n)/(1+ |x|) .

Furthermore, choosing an appropriate constant C, by Lemma 3 for every k > 1 we get

P[ |S*I > 1*1«] < C|x|-“ < Ce(n)/(1 + |x|). . and

P[ max |Si| >|x|« > S,] < c«-- £)/«“ |x|°

< G|x|-° + Cc1/2(n)|x|“-1/(l +|x|°) < Ce^OOfll+ |x|).

where

Z(fc,n,x) = {» > 1: tf(l - c2) < <“ < «“(1 + e,/2(n)|xr-‘)} . Hence

P[ISnJ> IxkrJ< C|x|-° < C?'2(n)/(1 + |x|) what with(46) gives (23). Inequalities(24) and(25) follows from .(23) and

P[ |5n.| > |*|<N./(1 - c2)*/o] v P[ ISjv.I > C|x|«<,/«.] <

< P[ |SN.| >J«kT„] < C|x|-° <Ce^n)/^ + |x|) .

(14)

56 ^A.^Krajka^,^Z. ^Rychlik

The first inequality is a consequence of the definitionofT„, i.e., <t. < and(21).

By (21) weget <,T, < <NB(l-C2),/o provided l-c2 < < 1+CiC1/2(n)|x|"-1.

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ZakladNauk Pomocniczych Instytut MatematykiUMCS Historii i BibliotekoznawstwaUMCS PlacM. Curie Sklodowskiej 1 Plac M. CurieSklodowskiej 1 20-031 Lublin,Poland 20-031 Lublin,Poland

(received April15, 1993)